Variance-gamma distribution

variance-gamma distribution

Parameters	$\mu$ location (real) $\alpha$ (real) $\beta$ asymmetry parameter (real) $\lambda > 0$ $\gamma = \sqrt{\alpha^2 - \beta^2} > 0$
Support	$x \in (-\infty; +\infty)\!$
PDF	$\frac{\gamma^{2\lambda} | x - \mu|^{\lambda-1/2} K_{\lambda-1/2} \left(\alpha|x - \mu|\right)}{\sqrt{\pi} \Gamma (\lambda)(2 \alpha)^{\lambda-1/2}} \; e^{\beta (x - \mu)}$ $K_\lambda$ denotes a modified Bessel function of the second kind $\Gamma$ denotes the Gamma function
Mean	$\mu + 2 \beta \lambda/ \gamma^2$
Variance	$2\lambda(1 + 2 \beta^2/\gamma^2)/\gamma^2$
MGF	$e^{\mu z} \left(\gamma/\sqrt{\alpha^2 -(\beta+z)^2}\right)^{2\lambda}$

The variance-gamma distribution, generalized Laplace distribution^[1] or Bessel function distribution^[1] is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta.^[2] The variance-gamma distributions form a subclass of the generalised hyperbolic distributions.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of variance-gamma distributions is closed under convolution in the following sense. If $X_1$ and $X_2$ are independent random variables that are variance-gamma distributed with the same values of the parameters $\alpha$ and $\beta$, but possibly different values of the other parameters, $\lambda_1$, $\mu_1$ and $\lambda_2,$ $\mu_2$, respectively, then $X_1 + X_2$ is variance-gamma distributed with parameters $\alpha$, $\beta$, $\lambda_1+\lambda_2$ and $\mu_1 + \mu_2$.

The variance-gamma distribution can also be expressed in terms of three inputs parameters (C,G,M) denoted after the initials of its founders. If the "C", $\lambda$ here, parameter is integer then the distribution has a closed form 2-EPT distribution. See 2-EPT Probability Density Function. Under this restriction closed form option prices can be derived.

See also Variance gamma process.

Differential equation

The pdf of the variance-gamma distribution is a solution of the following differential equation for $x>u$:

$\left\{\begin{array}{l} (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta (\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt] f(0)=\frac{\sqrt{\alpha} \left(-\frac{1}{2}\right)^{\lambda-\frac{1}{2}} e^{-\beta\mu} \mu^{\lambda-\frac{1}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} K_{\lambda-\frac{1}{2}}(-\alpha\mu)}{\sqrt{\pi} \Gamma(\lambda)}, \\[12pt] f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda} \mu e^{-\beta\mu} (-\mu )^{\lambda- \frac{5}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} \left((\beta\mu-2 \lambda+1) K_{\lambda -\frac{1}{2}}(-\alpha\mu)- \alpha\mu K_{\lambda+\frac{1}{2}}(-\alpha\mu)\right)}{\sqrt{\pi} \Gamma(\lambda)} \end{array}\right\}$

It is a solution of the following differential equation for $x<u$:

$\left\{\begin{array}{l} (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta(\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt] f(0)=\frac{2^{\frac{1}{2}-\lambda}\sqrt{\frac{\alpha }{\mu}} e^{-\beta\mu} \left(\mu \left(\alpha- \frac{\beta^2}{\alpha}\right)\right)^{\lambda} K_{\lambda-\frac{1}{2}}(\alpha\mu)}{\sqrt{\pi} \Gamma(\lambda)}, \\[12pt] f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda} e^{-\beta\mu} \mu^{\lambda-\frac{3}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} \left((\beta\mu-2 \lambda+1) K_{\lambda-\frac{1}{2}}(\alpha\mu)+\alpha\mu K_{\lambda+\frac{1}{2}}(\alpha\mu)\right)}{\sqrt{\pi} \Gamma (\lambda)} \end{array}\right\}$

Notes

1. Kotz, S.; et al. (2001). The Laplace Distribution and Generalizations. Birkhäuser. p. 180. ISBN 0-8176-4166-1.
2. D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, Journal of Business, 63, pp. 511–524.